604 CHAPTER 9 Polar Coordinates; Vectors How to Convert from Rectangular Coordinates to Polar Coordinates Where the Point Lies on a Coordinate Axis Find polar coordinates of a point whose rectangular coordinates are ( ) 0, 3 . Step-by-Step Solution Step 1 Plot the point x, y ( ) and note the quadrant the point lies in or the coordinate axis the point lies on. The point ( ) 0, 3 lies on the y-axis a distance of 3 units from the origin (pole), so = r 3. Plot the point ( ) 0, 3 in a rectangular coordinate system. See Figure 15. The point lies on the positive y-axis. Step 2 Find the distance r from the origin to the point. Step 3 Determine θ. Figure 15 (x, y) 5 (0, 3) x y 3 p– 2 Figure 14 verifies the result obtained in Example 4(a) using a TI-84 Plus CE. Note that the calculator is in radian mode. Now Work PROBLEMS 43 AND 55 3 Convert from Rectangular Coordinates to Polar Coordinates Converting from rectangular coordinates ( ) x y , to polar coordinates θ ( ) r, is a little more complicated. Notice that each solution begins by plotting the given rectangular coordinates. EXAMPLE 5 Most graphing calculators have the capability of converting from rectangular coordinates to polar coordinates. Consult your user’s manual for the proper keystrokes. Figure 16 verifies the results obtained in Example 5 using a TI-84 Plus CE. Note that the calculator is in radian mode. Figure 17 shows polar coordinates of points that lie on either the x-axis or the y-axis. In each illustration, > a 0. A ray with vertex at the pole through ( ) 0, 3 forms an angle θ π = 2 with the polar axis. Polar coordinates for this point can be given by π ( ) 3, 2 . Other possible representations include π ( ) − − 3, 2 and π ( ) 3, 5 2 . Figure 14 Figure 16 Figure 17 x (r, u) 5 (a, 0) (x, y) 5 (a, 0) a y (a) (x, y) 5 (a, 0), a . 0 p– 2 p– 2 x (r, u) 5 a, (x, y) 5 (0, a) a y (b) (x, y) 5 (0, a), a . 0 ( ) x (r, u) 5 (a, p) (x, y) 5 (2a, 0) p a y (c) (x, y) 5 ( 2a, 0), a . 0 (r, u) 5 a, x a y (d) (x, y) 5 (0, 2a), a . 0 (x, y) 5 (0, 2a) 3p–– 2 3p ––– 2 ( ) Now Work PROBLEM 59
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